The actual concept of a New Covenant actually emerges in the Hebrew Scriptures with the prophet Jeremiah-
Jeremiah 31:
31. Behold, the days come, saith the Lord, that I will make a new covenant with the house of Israel, and with the house of Judah:
32. Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the Lord:
33. But this shall be the covenant that I will make with the house of Israel; After those days, saith the Lord, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
34. And they shall teach no more every man his neighbour, and every man his brother, saying, Know the Lord: for they shall all know me, from the least of them unto the greatest of them, saith the Lord: for I will forgive their iniquity, and I will remember their sin no more.
35. Thus saith the Lord, which giveth the sun for a light by day, and the ordinances of the moon and of the stars for a light by night, which divideth the sea when the waves thereof roar; The Lord of hosts is his name:
36. If those ordinances depart from before me, saith the Lord, then the seed of Israel also shall cease from being a nation before me for ever.
37. Thus saith the Lord; If heaven above can be measured, and the foundations of the earth searched out beneath, I will also cast off all the seed of Israel for all that they have done, saith the Lord.
Of course, Jews & C’tians disagree on how this NC applies. Interestingly, Jeremiah says it will be established with Israel & Judah, no mention of the Gentiles (there is a C’tian explanation to this of which a lot of C’tians aren’t aware.)
Re the applicable laws- most Christians would agree that the Ten Commandments apply (with differing interpretations on the Sabbath & Sunday)- Jesus & the Apostles certainly repeat them enough. Also, Acts 14 & 15 reinforce what are known in Judaism as the Noachic Laws, God’s basic law for all humanity- against idolatry, blasphemy, murder, illicit sex acts, theft, cruelty to animals/eating blood, and for establishing courts of justice. Laws of animal sacrifice are totally out (a VERY few C’tians differ on this but the Letter to the Hebrews seems to maintain this). Various OT festivals & Kosher food laws are held by Paul in Romans 12 & Galatians to be matters of personal conviction.
AFAIK, Orthodox Jews must follow all laws which can be followed. The absence of the Altar of Sacrifice & an active Priesthood exempts them from the Sacrificial Laws. Also, AFAIK, they haven’t worked out a way to implement the Sabbath-year & Jubilee laws.